Abstract

We demonstrate quantitative phase delay measurements with a spiral phase contrast microscope working in confocal mode. Such a confocal configuration is sensitive to weak phase objects due to background rejection but does not give direct access to the phase delay introduced by the sample. We develop a theory showing that shifting the illumination spot relative to the detector gives access to the local phase gradient in the first-order approximation. Subsequently, we present an iterative integration algorithm for phase delay measurements. This approach is validated on simulated and calibrated experimental images. Finally, the algorithm is applied to measure the phase profile of a cell, in which phase delays of 10 mrad are observed.

© 2014 Optical Society of America

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References

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  1. R. Barer, “Interference microscopy and mass determination,” Nature 169, 366–367 (1952).
    [CrossRef]
  2. P. Jourdain, N. Pavillon, C. Moratal, D. Boss, B. Rappaz, C. Depeursinge, P. Marquet, and P. J. Magistretti, “Determination of transmembrane water fluxes in neurons elicited by glutamate ionotropic receptors and by the cotransporters KCC2 and NKCC1: a digital holographic microscopy study,” J. Neurosci. 31, 11846–11854 (2011).
    [CrossRef]
  3. A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Spiral interferogram analysis,” J. Opt. Soc. Am. A 23, 1400–1409 (2006).
    [CrossRef]
  4. G. Situ, M. Warber, G. Pedrini, and W. Osten, “Phase contrast enhancement in microscopy using spiral phase filtering,” Opt. Commun. 283, 1273–1277 (2010).
    [CrossRef]
  5. M. A. Lauterbach, M. Guillon, A. Soltani, and V. Emiliani, “STED microscope with spiral phase contrast,” Sci. Rep. 3, 2050 (2013).
    [CrossRef]
  6. C. Maurer, S. Bernet, and M. Ritsch-Marte, “Refining common path interferometry with a spiral phase Fourier filter,” J. Opt. A 11, 094023 (2009).
    [CrossRef]
  7. A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
    [CrossRef]
  8. S. Bernet, A. Jesacher, S. Fürhapter, C. Maurer, and M. Ritsch-Marte, “Quantitative imaging of complex samples by spiral phase contrast microscopy,” Opt. Express 14, 3792–3805 (2006).
    [CrossRef]
  9. D. K. Hamilton and T. Wilson, “Two-dimensional phase imaging in the scanning optical microscope,” Appl. Opt. 23, 348–352 (1984).
    [CrossRef]
  10. M. R. Atkinson and A. E. Dixon, “Single-pinhole confocal differential phase contrast microscopy,” Appl. Opt. 33, 641–653 (1994).
    [CrossRef]
  11. W. B. Amos, S. Reichelt, D. M. Cattermole, and J. Laufer, “Re-evaluation of differential phase contrast (DPC) in a scanning laser microscope using a split detector as an alternative to differential interference contrast (DIC) optics,” J. Microsc. 210, 166–175 (2003).
    [CrossRef]
  12. G. Brakenhoff, “Imaging modes in confocal scanning light-microscopy (CSLM),” J. Microsc. 117, 233–242 (1979).
    [CrossRef]
  13. C. Cogswell and C. Sheppard, “Confocal differential interference contrast (DIC) microscopy—including a theoretical-analysis of conventional and confocal DIC imaging,” J. Microsc. 165, 81–101 (1992).
    [CrossRef]
  14. N. Lue, W. Choi, G. Popescu, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Synthetic aperture tomographic phase microscopy for 3D imaging of live cells in translational motion,” Opt. Express 16, 16240 (2008).
    [CrossRef]
  15. N. Lue, W. Choi, K. Badizadegan, R. R. Dasari, M. S. Feld, and G. Popescu, “Confocal diffraction phase microscopy of live cells,” Opt. Lett. 33, 2074–2076 (2008).
    [CrossRef]
  16. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  17. E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. 24, 291–293 (1999).
    [CrossRef]
  18. C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
    [CrossRef]
  19. R. Gase, “Representation of Laguerre–Gaussian modes by the Wigner distribution function,” IEEE J. Quantum Electron. 31, 1811–1818 (1995).
    [CrossRef]
  20. I. Kimel and L. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
    [CrossRef]
  21. S.-C. Pei and C.-L. Liu, “A general form of 2D Fourier transform eigenfunctions,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (IEEE, 2012), pp. 3701–3704.
  22. P. Török and P. R. T. Munro, “The use of Gauss–Laguerre vector beams in STED microscopy,” Opt. Express 12, 3605–3617 (2004).
    [CrossRef]
  23. E. Wolf, “Three-dimensional structure determination of semitransparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
    [CrossRef]
  24. A. Devaney, “Inverse-scattering theory within the Rytov approximation,” Opt. Lett. 6, 374–376 (1981).
    [CrossRef]
  25. J. Pawley, ed., Confocal Microscopy with Transmitted Light (Plenum, 1995).

2013 (1)

M. A. Lauterbach, M. Guillon, A. Soltani, and V. Emiliani, “STED microscope with spiral phase contrast,” Sci. Rep. 3, 2050 (2013).
[CrossRef]

2011 (2)

P. Jourdain, N. Pavillon, C. Moratal, D. Boss, B. Rappaz, C. Depeursinge, P. Marquet, and P. J. Magistretti, “Determination of transmembrane water fluxes in neurons elicited by glutamate ionotropic receptors and by the cotransporters KCC2 and NKCC1: a digital holographic microscopy study,” J. Neurosci. 31, 11846–11854 (2011).
[CrossRef]

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
[CrossRef]

2010 (1)

G. Situ, M. Warber, G. Pedrini, and W. Osten, “Phase contrast enhancement in microscopy using spiral phase filtering,” Opt. Commun. 283, 1273–1277 (2010).
[CrossRef]

2009 (1)

C. Maurer, S. Bernet, and M. Ritsch-Marte, “Refining common path interferometry with a spiral phase Fourier filter,” J. Opt. A 11, 094023 (2009).
[CrossRef]

2008 (2)

2006 (2)

2005 (1)

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[CrossRef]

2004 (1)

2003 (1)

W. B. Amos, S. Reichelt, D. M. Cattermole, and J. Laufer, “Re-evaluation of differential phase contrast (DPC) in a scanning laser microscope using a split detector as an alternative to differential interference contrast (DIC) optics,” J. Microsc. 210, 166–175 (2003).
[CrossRef]

1999 (1)

1995 (1)

R. Gase, “Representation of Laguerre–Gaussian modes by the Wigner distribution function,” IEEE J. Quantum Electron. 31, 1811–1818 (1995).
[CrossRef]

1994 (1)

1993 (1)

I. Kimel and L. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
[CrossRef]

1992 (1)

C. Cogswell and C. Sheppard, “Confocal differential interference contrast (DIC) microscopy—including a theoretical-analysis of conventional and confocal DIC imaging,” J. Microsc. 165, 81–101 (1992).
[CrossRef]

1984 (1)

1981 (1)

1979 (1)

G. Brakenhoff, “Imaging modes in confocal scanning light-microscopy (CSLM),” J. Microsc. 117, 233–242 (1979).
[CrossRef]

1969 (1)

E. Wolf, “Three-dimensional structure determination of semitransparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

1952 (1)

R. Barer, “Interference microscopy and mass determination,” Nature 169, 366–367 (1952).
[CrossRef]

Amos, W. B.

W. B. Amos, S. Reichelt, D. M. Cattermole, and J. Laufer, “Re-evaluation of differential phase contrast (DPC) in a scanning laser microscope using a split detector as an alternative to differential interference contrast (DIC) optics,” J. Microsc. 210, 166–175 (2003).
[CrossRef]

Atkinson, M. R.

Badizadegan, K.

Barer, R.

R. Barer, “Interference microscopy and mass determination,” Nature 169, 366–367 (1952).
[CrossRef]

Bernet, S.

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
[CrossRef]

C. Maurer, S. Bernet, and M. Ritsch-Marte, “Refining common path interferometry with a spiral phase Fourier filter,” J. Opt. A 11, 094023 (2009).
[CrossRef]

S. Bernet, A. Jesacher, S. Fürhapter, C. Maurer, and M. Ritsch-Marte, “Quantitative imaging of complex samples by spiral phase contrast microscopy,” Opt. Express 14, 3792–3805 (2006).
[CrossRef]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Spiral interferogram analysis,” J. Opt. Soc. Am. A 23, 1400–1409 (2006).
[CrossRef]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[CrossRef]

Bevilacqua, F.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Boss, D.

P. Jourdain, N. Pavillon, C. Moratal, D. Boss, B. Rappaz, C. Depeursinge, P. Marquet, and P. J. Magistretti, “Determination of transmembrane water fluxes in neurons elicited by glutamate ionotropic receptors and by the cotransporters KCC2 and NKCC1: a digital holographic microscopy study,” J. Neurosci. 31, 11846–11854 (2011).
[CrossRef]

Brakenhoff, G.

G. Brakenhoff, “Imaging modes in confocal scanning light-microscopy (CSLM),” J. Microsc. 117, 233–242 (1979).
[CrossRef]

Cattermole, D. M.

W. B. Amos, S. Reichelt, D. M. Cattermole, and J. Laufer, “Re-evaluation of differential phase contrast (DPC) in a scanning laser microscope using a split detector as an alternative to differential interference contrast (DIC) optics,” J. Microsc. 210, 166–175 (2003).
[CrossRef]

Choi, W.

Cogswell, C.

C. Cogswell and C. Sheppard, “Confocal differential interference contrast (DIC) microscopy—including a theoretical-analysis of conventional and confocal DIC imaging,” J. Microsc. 165, 81–101 (1992).
[CrossRef]

Cuche, E.

Dasari, R. R.

Depeursinge, C.

P. Jourdain, N. Pavillon, C. Moratal, D. Boss, B. Rappaz, C. Depeursinge, P. Marquet, and P. J. Magistretti, “Determination of transmembrane water fluxes in neurons elicited by glutamate ionotropic receptors and by the cotransporters KCC2 and NKCC1: a digital holographic microscopy study,” J. Neurosci. 31, 11846–11854 (2011).
[CrossRef]

E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. 24, 291–293 (1999).
[CrossRef]

Devaney, A.

Dixon, A. E.

Elias, L.

I. Kimel and L. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
[CrossRef]

Emiliani, V.

M. A. Lauterbach, M. Guillon, A. Soltani, and V. Emiliani, “STED microscope with spiral phase contrast,” Sci. Rep. 3, 2050 (2013).
[CrossRef]

Feld, M. S.

Fürhapter, S.

Gase, R.

R. Gase, “Representation of Laguerre–Gaussian modes by the Wigner distribution function,” IEEE J. Quantum Electron. 31, 1811–1818 (1995).
[CrossRef]

Guillon, M.

M. A. Lauterbach, M. Guillon, A. Soltani, and V. Emiliani, “STED microscope with spiral phase contrast,” Sci. Rep. 3, 2050 (2013).
[CrossRef]

Hamilton, D. K.

Jesacher, A.

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
[CrossRef]

S. Bernet, A. Jesacher, S. Fürhapter, C. Maurer, and M. Ritsch-Marte, “Quantitative imaging of complex samples by spiral phase contrast microscopy,” Opt. Express 14, 3792–3805 (2006).
[CrossRef]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Spiral interferogram analysis,” J. Opt. Soc. Am. A 23, 1400–1409 (2006).
[CrossRef]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[CrossRef]

Jourdain, P.

P. Jourdain, N. Pavillon, C. Moratal, D. Boss, B. Rappaz, C. Depeursinge, P. Marquet, and P. J. Magistretti, “Determination of transmembrane water fluxes in neurons elicited by glutamate ionotropic receptors and by the cotransporters KCC2 and NKCC1: a digital holographic microscopy study,” J. Neurosci. 31, 11846–11854 (2011).
[CrossRef]

Kimel, I.

I. Kimel and L. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
[CrossRef]

Laufer, J.

W. B. Amos, S. Reichelt, D. M. Cattermole, and J. Laufer, “Re-evaluation of differential phase contrast (DPC) in a scanning laser microscope using a split detector as an alternative to differential interference contrast (DIC) optics,” J. Microsc. 210, 166–175 (2003).
[CrossRef]

Lauterbach, M. A.

M. A. Lauterbach, M. Guillon, A. Soltani, and V. Emiliani, “STED microscope with spiral phase contrast,” Sci. Rep. 3, 2050 (2013).
[CrossRef]

Liu, C.-L.

S.-C. Pei and C.-L. Liu, “A general form of 2D Fourier transform eigenfunctions,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (IEEE, 2012), pp. 3701–3704.

Lue, N.

Magistretti, P. J.

P. Jourdain, N. Pavillon, C. Moratal, D. Boss, B. Rappaz, C. Depeursinge, P. Marquet, and P. J. Magistretti, “Determination of transmembrane water fluxes in neurons elicited by glutamate ionotropic receptors and by the cotransporters KCC2 and NKCC1: a digital holographic microscopy study,” J. Neurosci. 31, 11846–11854 (2011).
[CrossRef]

Marquet, P.

P. Jourdain, N. Pavillon, C. Moratal, D. Boss, B. Rappaz, C. Depeursinge, P. Marquet, and P. J. Magistretti, “Determination of transmembrane water fluxes in neurons elicited by glutamate ionotropic receptors and by the cotransporters KCC2 and NKCC1: a digital holographic microscopy study,” J. Neurosci. 31, 11846–11854 (2011).
[CrossRef]

Maurer, C.

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
[CrossRef]

C. Maurer, S. Bernet, and M. Ritsch-Marte, “Refining common path interferometry with a spiral phase Fourier filter,” J. Opt. A 11, 094023 (2009).
[CrossRef]

S. Bernet, A. Jesacher, S. Fürhapter, C. Maurer, and M. Ritsch-Marte, “Quantitative imaging of complex samples by spiral phase contrast microscopy,” Opt. Express 14, 3792–3805 (2006).
[CrossRef]

Moratal, C.

P. Jourdain, N. Pavillon, C. Moratal, D. Boss, B. Rappaz, C. Depeursinge, P. Marquet, and P. J. Magistretti, “Determination of transmembrane water fluxes in neurons elicited by glutamate ionotropic receptors and by the cotransporters KCC2 and NKCC1: a digital holographic microscopy study,” J. Neurosci. 31, 11846–11854 (2011).
[CrossRef]

Munro, P. R. T.

Osten, W.

G. Situ, M. Warber, G. Pedrini, and W. Osten, “Phase contrast enhancement in microscopy using spiral phase filtering,” Opt. Commun. 283, 1273–1277 (2010).
[CrossRef]

Pavillon, N.

P. Jourdain, N. Pavillon, C. Moratal, D. Boss, B. Rappaz, C. Depeursinge, P. Marquet, and P. J. Magistretti, “Determination of transmembrane water fluxes in neurons elicited by glutamate ionotropic receptors and by the cotransporters KCC2 and NKCC1: a digital holographic microscopy study,” J. Neurosci. 31, 11846–11854 (2011).
[CrossRef]

Pedrini, G.

G. Situ, M. Warber, G. Pedrini, and W. Osten, “Phase contrast enhancement in microscopy using spiral phase filtering,” Opt. Commun. 283, 1273–1277 (2010).
[CrossRef]

Pei, S.-C.

S.-C. Pei and C.-L. Liu, “A general form of 2D Fourier transform eigenfunctions,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (IEEE, 2012), pp. 3701–3704.

Popescu, G.

Rappaz, B.

P. Jourdain, N. Pavillon, C. Moratal, D. Boss, B. Rappaz, C. Depeursinge, P. Marquet, and P. J. Magistretti, “Determination of transmembrane water fluxes in neurons elicited by glutamate ionotropic receptors and by the cotransporters KCC2 and NKCC1: a digital holographic microscopy study,” J. Neurosci. 31, 11846–11854 (2011).
[CrossRef]

Reichelt, S.

W. B. Amos, S. Reichelt, D. M. Cattermole, and J. Laufer, “Re-evaluation of differential phase contrast (DPC) in a scanning laser microscope using a split detector as an alternative to differential interference contrast (DIC) optics,” J. Microsc. 210, 166–175 (2003).
[CrossRef]

Ritsch-Marte, M.

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
[CrossRef]

C. Maurer, S. Bernet, and M. Ritsch-Marte, “Refining common path interferometry with a spiral phase Fourier filter,” J. Opt. A 11, 094023 (2009).
[CrossRef]

S. Bernet, A. Jesacher, S. Fürhapter, C. Maurer, and M. Ritsch-Marte, “Quantitative imaging of complex samples by spiral phase contrast microscopy,” Opt. Express 14, 3792–3805 (2006).
[CrossRef]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Spiral interferogram analysis,” J. Opt. Soc. Am. A 23, 1400–1409 (2006).
[CrossRef]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[CrossRef]

Sheppard, C.

C. Cogswell and C. Sheppard, “Confocal differential interference contrast (DIC) microscopy—including a theoretical-analysis of conventional and confocal DIC imaging,” J. Microsc. 165, 81–101 (1992).
[CrossRef]

Situ, G.

G. Situ, M. Warber, G. Pedrini, and W. Osten, “Phase contrast enhancement in microscopy using spiral phase filtering,” Opt. Commun. 283, 1273–1277 (2010).
[CrossRef]

Soltani, A.

M. A. Lauterbach, M. Guillon, A. Soltani, and V. Emiliani, “STED microscope with spiral phase contrast,” Sci. Rep. 3, 2050 (2013).
[CrossRef]

Török, P.

Warber, M.

G. Situ, M. Warber, G. Pedrini, and W. Osten, “Phase contrast enhancement in microscopy using spiral phase filtering,” Opt. Commun. 283, 1273–1277 (2010).
[CrossRef]

Wilson, T.

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semitransparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Appl. Opt. (2)

IEEE J. Quantum Electron. (2)

R. Gase, “Representation of Laguerre–Gaussian modes by the Wigner distribution function,” IEEE J. Quantum Electron. 31, 1811–1818 (1995).
[CrossRef]

I. Kimel and L. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
[CrossRef]

J. Microsc. (3)

W. B. Amos, S. Reichelt, D. M. Cattermole, and J. Laufer, “Re-evaluation of differential phase contrast (DPC) in a scanning laser microscope using a split detector as an alternative to differential interference contrast (DIC) optics,” J. Microsc. 210, 166–175 (2003).
[CrossRef]

G. Brakenhoff, “Imaging modes in confocal scanning light-microscopy (CSLM),” J. Microsc. 117, 233–242 (1979).
[CrossRef]

C. Cogswell and C. Sheppard, “Confocal differential interference contrast (DIC) microscopy—including a theoretical-analysis of conventional and confocal DIC imaging,” J. Microsc. 165, 81–101 (1992).
[CrossRef]

J. Neurosci. (1)

P. Jourdain, N. Pavillon, C. Moratal, D. Boss, B. Rappaz, C. Depeursinge, P. Marquet, and P. J. Magistretti, “Determination of transmembrane water fluxes in neurons elicited by glutamate ionotropic receptors and by the cotransporters KCC2 and NKCC1: a digital holographic microscopy study,” J. Neurosci. 31, 11846–11854 (2011).
[CrossRef]

J. Opt. A (1)

C. Maurer, S. Bernet, and M. Ritsch-Marte, “Refining common path interferometry with a spiral phase Fourier filter,” J. Opt. A 11, 094023 (2009).
[CrossRef]

J. Opt. Soc. Am. A (1)

Laser Photon. Rev. (1)

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
[CrossRef]

Nature (1)

R. Barer, “Interference microscopy and mass determination,” Nature 169, 366–367 (1952).
[CrossRef]

Opt. Commun. (2)

G. Situ, M. Warber, G. Pedrini, and W. Osten, “Phase contrast enhancement in microscopy using spiral phase filtering,” Opt. Commun. 283, 1273–1277 (2010).
[CrossRef]

E. Wolf, “Three-dimensional structure determination of semitransparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. Lett. (1)

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[CrossRef]

Sci. Rep. (1)

M. A. Lauterbach, M. Guillon, A. Soltani, and V. Emiliani, “STED microscope with spiral phase contrast,” Sci. Rep. 3, 2050 (2013).
[CrossRef]

Other (3)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

S.-C. Pei and C.-L. Liu, “A general form of 2D Fourier transform eigenfunctions,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (IEEE, 2012), pp. 3701–3704.

J. Pawley, ed., Confocal Microscopy with Transmitted Light (Plenum, 1995).

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Figures (8)

Fig. 1.
Fig. 1.

(a) Principle of confocal scanning SPC imaging. An illumination source (electric field Ei) is imaged onto the sample object O, which in turn is imaged onto a detection pinhole. In the Fourier plane of the imaging objective lens, light passes through a mask that introduces a helical phase delay and an amplitude profile Π. An image is recorded by scanning the object O relative to the illumination spot and the detection pinhole. (b) Illustration of the helical phase mask introducing a phase delay equal to the azimuthal coordinate ϕ. Phase in radians is represented with linear gray levels between 0 and 2π.

Fig. 2.
Fig. 2.

Numerical simulations showing the real part of the FFAPSF of SPC with an oil 1.4 NA objective lens and a wavelength of 640 nm. Gray solid curve, FFAPSF for a circular top-hat pupil; green-dashed curve, FFAPSF for a Gaussian-shaped pupil; red dotted curve, apodized FFAPSF for a LG pupil function LG01.

Fig. 3.
Fig. 3.

Validation of algorithms on simulated data. Gray solid curves: theoretical phase delay introduced by a 1.3 μm bead with an index contrast of δn=4·102. Pupil profiles are simulated to be (a) a circular top-hat with helical phase and (b) a LG01 function. Green dashed curves: phase obtained by direct integration of simulated CSPC images. Black dotted curves: phase reconstruction obtained using the iterative algorithm described in Section 3.C. Convergence is obtained after (a) 50 iterations and (b) only 5 iterations. Numerical simulations for a 1.4 NA objective lens, wavelength λ=640nm.

Fig. 4.
Fig. 4.

Iterative algorithm for phase retrieval. (a) Schematic diagram of the iterative algorithm converging toward the phase profile Θ of the sample. The main operator J involved in the iterative loop is made explicit in (b). To find an approximation Θguess of the true phase profile Θ, a set of correcting images C (successively approximated by Ck) is searched to be added to the normalized experimental difference images Γd, so that Step 1 of operator J gives the estimate Θguess of the phase map Θ. If the proper correcting images C are found, then Γk=Γd. Otherwise, Θguess is successively approximated by Θk, and the algorithm is run until ΓdΓk<ϵ, where ϵ is the convergence criterion.

Fig. 5.
Fig. 5.

Comparison of quantitative CSPC and AFM images on polymerized epoxy droplets embedded in water. The height is (a) reconstructed by quantitative CSPC and (b) measured by AFM. Since the sample was imaged with two different instruments, the scale and orientation are slightly different. For quantitative CSPC, the refractive index contrast Δn=1.541.33=0.21 is obtained from the epoxy manufacturer and the tabulated value for water. In image (b), arrows indicate the line along which a height profile is plotted in Fig. 6.

Fig. 6.
Fig. 6.

Quantitative phase retrieval validated on epoxy droplets. The relative difference of the height profile between the AFM measurement (gray solid line) and the quantitative CSPC measurement (green dashed line) is of the order of 15%.

Fig. 7.
Fig. 7.

Phase retrieval in biological samples. Images of an astrocyte embedded in an index-matched mounting medium. The illumination was (a) centered or (b) and (c) in two orthogonal directions slightly shifted with respect to the detection pinhole. In the images (b) and (c), phase gradients along (b) horizontal and (c) vertical directions are highlighted. Image (d) shows the quantitative phase reconstruction obtained by the iterative algorithm. Gray scale is in milliradians and scale bar is 10 μm. The phase profile along the white dotted line is plotted in figure (e), showing that the filopodium that is clearly resolved in (d) introduces a phase delay of the order of only 10 mrad.

Fig. 8.
Fig. 8.

Suppressing out-of-focus objects with CSPC. Silica beads of 2.34 μm diameter in water are lying in two different planes. (a) and (b) images of these two planes with CSPC; (c) and (d) scanning SPC with collimated laser illumination. (a) and (c) images of two beads 75μm above the plane of the large bead clusters presented beneath (b) and (d). With widefield illumination of the sample, the image of the two beads is unreadable because the bead clusters below are not suppressed. Scale bar is 5 μm.

Equations (64)

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Ed0(r)=D(r)Ei(rr)d2r,
D(r)=12πΠ(k)exp(iϕ)exp(ik·r)d2k.
Ed(r0,r)=D(r)Ei(rr)O(r0r)d2r.
F(LGpl)=(i)(2p+l)LGpl,
LG01(k)=π1/2exp(iϕ)|k|exp[k2/(4α)],
O(r0)=exp[iΘ(r0)],
Ed(r0,r)=D(r)Ei(rr)exp[iΘ(r0r)]d2r.
exp[iΘ(r0r)]=exp[iΘ(r0)]exp[iΔΘ(r0,r)],
Ed(r0,r)=exp[iΘ(r0)][D(r)Ei(rr)d2r+iD(r)Ei(rr)ΔΘ(r0,r)d2r].
ΔΘ(r0,r)=Θ(r0r)Θ(r0),
r·Θ(r0),
Ed(r0,r)=exp[iΘ(r0)][R(r)iΘ(r0)·S(r)],
R(r)=D(r)Ei(rr)d2r,
S(r)=rD(r)Ei(rr)d2r.
Id(r0,r)=|R(r)iS(r)·Θ(r0)|2,
=|R(r)|2+|S(r)·Θ(r0)|2[iR*(r)S(r)·Θ(r0)+cc],
R=2αr·S.
S=iπ3/2α3/22α2(ux+iuy),
Ed(r0,r)=iπ3/2α3/22α2exp[iΘ(r0)][2αr¯iΘ¯(r0)],
Id=π3(αα)3[αr2+r|Θ|sin(φΘφ)+|Θ|24α],
Id,c=π3(αα)3|Θ|24α.
Γx=Id,xId,c=π3(αα)3r(αr+xΘ),
Γy=Id,yId,c=π3(αα)3r(αr+yΘ).
Γx=1+xΘαr,
Γy=1+yΘαr.
F(Θ¯)=F(xΘ+iyΘ),
=ik¯F(Θ),
Θ=iαrF1{k¯1F[(Γx1)+i(Γy1)]}.
Γx=Id,xId,cId,x|Θ=0,
Γy=Id,yId,cId,y|Θ=0.
J(Γd+C)=Γd.
Γk=J(Γd+Ck),
Ck+1=Ck+(ΓdΓk).
R(r)=G(rr)D(r)d2r.
R(r)=12πG(rr)[Π(k)exp(iϕ)exp(ik.r)d2k]d2r
=12πΠ(k)exp(iϕ)exp(ik.r)[G(r)exp(ik.r)d2r]d2k
=12πΠ(k)exp(iϕ)exp(ik.r)F(G)(k)d2k,
F(G)(k)=παexp(k24α).
Π(k)=exp(k24αim).
Π(k)×F(G)(k)=παexp(k24α),
α1=α1+αim1.
R(r)=12αexp(k24α+iϕ+ik·r)d2k,
R(r)=2ααexp(k2+iϕ+2iαk·r)d2k.
R(r)=2αα[exp(k2+iϕ)d2k+2iαr·kexp(k2+iϕ)d2k]
=4iα3/2αr·kexp(k2+iϕ)d2k,
R(r)=iπ3/2α3/2αr·(ux+iuy).
S(r)=rG(rr)D(r)d2r
=12πrG(rr)[Π(k)exp(iϕ)exp(ik·r)d2k]d2r
=12πΠ(k)exp(iϕ)[rG(rr)exp(ik·r)d2r]d2k
=12πΠ(k)exp(iϕ)exp(ik·r)[(rr)G(r)exp(ik·r)d2r]d2k.
12πΠ(k)exp(iϕ)exp(ik·r)[rG(r)exp(ik·r)d2r]d2k=rR(r).
12π(r)G(r)exp(ik·r)d2r=ik[12πG(r)exp(ik·r)d2r]
=i4α2kexp(k24α).
S(r)=rR(r)+i4α2kexp(k24α+iϕ+ik·r)d2k=rR(r)+12αrR.
S=12αrR.
R=2αr·S
S(r)=iπ3/22α3/2α2(ux+iuy).
kexp(αkk2)exp(iϕ)d2k,
kexp(αkk2)exp(iϕ)d2k=ukexp(iϕ)[k2exp(αkk2)dk]dϕ
=[ukexp(iϕ)dϕ][k2exp(αkk2)dk].
uk=12[exp(iϕ)(uxiuy)+exp(iϕ)(ux+iuy)],
ukexp(iϕ)dϕ=π(ux+iuy).
k2exp(αkk2)dk=αk[exp(αkk2)dk]=π4αk3/2.
kexp(αkk2)exp(iϕ)d2k=π3/24αk3/2(ux+iuy).

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